# Chapter 11/12 Theorems/Postulates/ Collaries

Theorem/Postulate/Corollary Definition
Area of a Square postulate the area of a square is the square of the length of its side, or A=s squared
Area Congruence postulate if two polyhedrons are congruent, then they have the same area
Area Addition postulate the area of a region is the sum of the areas of all its overlapping parts
Volume of a Cube the volume of a cube is the cube of the length of its side, or V=s cubed
Volume Congruence postulate If two polyhedra are congruent, then they have the same volume
Volume Addition postulate the volume of a solid is the sum of the volumes of all its overlapping parts
Area of Rectangle the area of a rectangle is the product of its base and height; A=bh
Area of a Parallelogram the area of a parallelogram is the product of a base and its corresponding height; A=bh
Area of a Triangle the area of a triangle is one half the product of the base and its corresponding height; A=1/2bh
Area of a Trapezoid the area of a trapezoid is one half the product of the height and the sum of the lengths of the + bases; A=1/2h[b(1)+b(2)]
Area of a Rhombus the area of a rhombus is one half the product of the lengths of its diagonals; A=1/2d(1)x d(2)
Area of a Kite the area of a kite is one half the product of the lengths of its diagonals; A=1/2d(1)x d(2)
Area of Similar Polygons if two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a squared: b squared
Circumference of a Circle the circumference C of a circle is C = pie x d or C = 2 x pie x r, where d is the diameter of the circle and r is the radius of the circle
Arc Length Corollary in a circle, the ratio of the length of a given arc to the the circumference is equal to the ratio of the ratio of the measure of the arc to 360 degrees.
Area of a Circle the area of a circle is pie times the square of the radius; A = pie x r squared
Area of a Sector the ratio of area A of a sector of a circle to the area of the whole sector of a circle to the area of the whole circle (pie x r squared) is equal to the ratio of the measure of the intercepted arc to 360 degrees; A/pie x r squared = m arc AB/360 degrees,
Area of a Regular Polygon the area of a regular n-gon with side length s is half the product of the apothem a and the perimeter; P, so A = 1/2aP, or A=1/2a x ns
Euler's Theorem the number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F + V = E + 2
Surface Area of a Right Prism the surface area S of a right prism is S = 2B + Ph = aP + Ph, where is the apothem of the base, B is the area of a base, P is the perimeter of a base, and h is the height
Surface Area of a Right Cylinder the surface area S of a right cylinder is S = 2B + Ch = 2 x pie x r squares + 2 x pie x r x h, where B is the area of a base, C is the circumference of a base, r is the radius of a base, and h is the height
Surface Area of a Regular Pyramid the surface area S of a regular pyramid is S = B + 1/2Pl, where B is the area of the base, P is the perimeter of the base, and l is the slant height
Surface Area of a Right Cone the surface are S of a right cone is S = B + 1/2Cl = pie x r squared + pie x r x l, where B is the area of a the base, C is the circumference of the base, r is the radius of the base, and l is the slant height
Volume of Prism the volume V of a prism is V = Bh, where b is the area of a base and h is the height
Volume of a Cylinder the volume V of a cylinder is V=Bh=pie x r squared x h, where B is the area of a base, h is the height, and r is the radius of a base
Cavalieri's Principle if two solids have the same height and the same cross-sectional area at every level, then they have the same volume
Volume of a Pyramid the volume V of a pyramid is V = 1/3Bh = 1/3 x pie x r squared x h, where b is the area of the base, h is the height, and r is the radius of the base
Surface Area of a Sphere the surface area S of a sphere with radius r is S=4 x pie x r cubed
Volume of a Sphere the volume V of a sphere with radius r is V = 4/3 x pie x r
Similar Solids Theorem if two similar solids of a:b, then corresponding areas have a ratio of a squared:b squared, and corresponding volumes have a ratio of a cubed:b cubed